I am looking for examples of sequences of adjoint functors. That are (possibly bounded) sequences
$$(...,F_{-1}, F_{0}, F_1, F_2,...)$$
such that each $F_n$ is left adjoint to $F_{n+1}$. We call such a sequence cyclic of order $k$ if for one $n$ (and hence for all) we have $F_{n} \cong F_{n+k}$.
It is relatively easy to prove that cyclic sequences of all orders and non-cyclic sequences of all possible length exist. This can e.g. be done using posets, see http://www.springerlink.com/content/pmj5074147116273/.

I am looing for more "natural" examples of such sequences that are as long as possible. By natural I mean that they grow out of "usual functors" (sorry for this vague statement...)

Let my give two short examples:
1) Let $U: Top \to Set$ be the forgetful functor from locally connected topological spaces to sets. This induces a sequence of length 4:
$$ (\pi_0 , Dis , U , CoDis) $$
where $Dis$ and $CoDis$ are the functor that equip a set with the discrete and indiscrete topology. Then the sequence stops. Tons of examples of this type are induced by pullback functors in algebraic geometry.

2) a cyclic sequence of order 2: the Diagaonal functor $\Delta: A \to A \times A$ for any abelian category $A$ is left and right adjoint to the direct sum
$$ ( ...,\Delta,\oplus,\Delta,\oplus,...)$$

8 Answers
8

The functor from the category of abelian groups to the category of arrows of abelian groups that sends an object to its identity morphism has three adjoints to the left and three to the right, for a chain of seven functors. The extreme adjoints are the functors that assign to an arrow its kernel or cokernel, as an object.

Pardon my ignorance, but what are the morphisms in the category of arrows?
–
Qiaochu YuanNov 22 '10 at 16:04

This is really a striking-simple example. I wonder that I have not come across it so far. Morphisms in the arrow-category are commuting squares.
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Thomas NikolausNov 22 '10 at 16:43

6

This isn't specific to Abelian groups, it works in any pointed category (i.e. a category with an object $0$ which both initial and final) having pullbacks and pushouts (where the kernel of a morphism $A \to B$ is the pullback of $A\to B \leftarrow 0$, and dually for the cokernel)
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Omar Antolín-CamarenaMay 8 '12 at 2:40

If $f:X \to Y$ is a proper morphism of algebraic varieties, and $D(X)$, $D(Y)$ are the (unbounded) derived categories of quasicoherent sheaves then $(f^*,f_*,f^!)$ is such a sequence of adjoint functors. If moreover, $f$ has finite Tor-dimension then $f^!(F) \cong f^* (F)\otimes f^!(O_Y)$. If moreover the relative dualizing complex $f^!(O_Y)$ is an invertible sheaf then the functor $T$ of tensoring with $f^!(O_Y)$ is an autoequivalence, hence we have an infinite sequence of adjoint functors
$$
(\dots,T^{-1}\circ f^*,f_*\circ T,f^*,f_*,T\circ f^*,f_*\circ T^{-1},T^2\circ f^*,f_*\circ T^{-2},\dots).
$$
The same happens for arbitrary pair of adjoint functors between categories which have Serre functors.

A nice one from the representation theory of $p$-adic reductive groups:

Let $k$ be a finite extension of $Q_p$. Let $G$ be the $k$-points of a connected reductive group over $k$. We do not distinguish between algebraic groups over $k$ and their $k$-points in what follows. Let $P$ be a parabolic $k$-subgroup of $G$. Let $M$ be a Levi subgroup of $P$, and $N$ the unipotent radical of $P$, so $P = MN$. Let $Q$ be the opposite parabolic to $P$, so that $Q \cap P = M$. Let $U$ be the unipotent radical of $Q$, so $Q = MU$.

Let $Rep(G)$ and $Rep(M)$ denote the categories of smooth representations of $G$ and $M$, respectively.

Let $R_P^G$ (respectively $R_Q^G$) denote Jacquet's restriction from $Rep(G)$ to $Rep(M)$, taking a representation $V$ of $G$ to its $N$-coinvariants $V_N$ (resp. $U$-coinvariants $V_U$), viewed as a representation of $M$.

Let [n] denote the totally ordered (n+1)-element set, regarded as a category. For each positive integer n, we have the usual n+1 order-preserving injections from [n-1] to [n], and the usual n order-preserving surjections from [n] to [n-1]. (I mean the ones used all the time for simplicial anything.) When you regard them as functors, these injections and surjections interleave to form an adjoint chain of length 2n.

I hadn't seen this question before, but this is almost a canonical answer. Just as $\Delta$ is the "generic monoid", you could call the 2-category $\Delta$ the "generic Kock-Zoeberlein monad", where the multiplication is left adjoint to a unit.
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Todd Trimble♦Feb 29 '12 at 2:59

3

Yeah, now that I think about it, this is probably one of the examples used in the paper Thomas cites. (I only just noticed that he talked about it being "done using posets".) But what's noticeable, I think, is that the functors involved come up naturally in non-categorical mathematics, e.g. the definition of singular homology.
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Tom LeinsterFeb 29 '12 at 3:08

@Tom: Thats exactly how they do it in the paper (of course with some modifications).
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Thomas NikolausMar 5 '12 at 2:27

Similar to Ben Wieland's and Sasha's answer: Let $\mathcal{C}$ be the category of complexes in an abelian category. Let $\underline{\mathcal{C}(\Delta_0)}$ be the homotopy category of $\mathcal{C}$. Let $\underline{\mathcal{C}(\Delta_1)}$ be the category of arrows with values in $\mathcal{C}$, with pointwise homotopy equivalences formally inverted (not to be confused with $\underline{\mathcal{C}(\Delta_0)}(\Delta_1)$). The functor $\underline{\mathcal{C}(\Delta_0)}\to\underline{\mathcal{C}(\Delta_1)}$ that sends an object of $\mathcal{C}$ to the identity arrow on this object, morphisms accordingly, gives rise to an infinite chain of adjoint functors ("walking along a distinguished triangle"). [Which functors, such as $\Delta_1\to\Delta_0$ considered here, have this property?]

A simple version of your first example is to look at the forgetful functor $U : \text{Graph} \to \text{Set}$, where $\text{Graph}$ is, say, the category of simple undirected graphs. (To be explicit, the morphisms in this category are maps of vertices which respect the edge relation, and in the edge relation a vertex is considered related to itself.) This functor has a left adjoint $E : \text{Set} \to \text{Graph}$ which sends a set to the empty graph on that set and a right adjoint $K : \text{Set} \to \text{Graph}$ which sends a set to the complete graph on that set. $K$ doesn't preserve coproducts, so it doesn't have a right adjoint. $E$ has a left adjoint $\pi_0 : \text{Graph} \to \text{Set}$ which sends a graph to its set of connected components (no topological difficulties here). I am not sure whether $\pi_0$ has a left adjoint in this case.

Here's an example similar in feel to the $\operatorname{Dis} \dashv U \dashv \operatorname{Codiss}$ example - so perhaps not of the sort that you were really after.

Let ${\bf Op}_1$ be the category whose objects are (complete) operator spaces (or "quantum/quantized Banach spaces" according to some authors) and whose morphisms are the completely contractive maps.
Let ${\bf Ban}_1$ be the category of Banach spaces and contractive (a.k.a. short) linear maps.
Then if $U:{\bf Op}_1\to{\bf Ban}_1$ is the forgetful functor, we have adjunctions $\operatorname{MAX} \dashv U \dashv \operatorname{MIN}$.

The left and right adjoints to $U$ are sometimes called the maximal and minimal quantizations, respectively, of a Banach space. (One also sees the terminology of "maximal and miimal operator space structures", but then we wouldn't be able to have the magic word quantum and its important-sounding derivatives.)